GRAPH A LINEAR INEQUALITY IN THE COORDINATE PLANE

Example 1 :

Sketch the graph of the following linear inequality.

y ≤ (3x/5) - 5

Solution :

Step 1 :

Consider the given inequality as equation.

y  =  (3x/5) - 5

The equation above is in slope-intercept form.

Slope  =  3/5

y-intercept  =  -5 

Using the slope and y-intercept, sketch the graph of the line. 

Step 2 :

Take a point on either sides of the line and check which point satisfies the given inequality. 

Step 3 :

From the above step, the point (5, -5) satisfies the inequality. 

So, we have to shade the region from where the point (5, -5) is taken. 

Example 2 :

Sketch the graph of the following linear inequality.

y > -x - 5

Solution :

As we have done in example 1, we can do steps 1, 2 and 3 and get the graph of the inequality as shown below. 

Example 3 :

An online store sells digital cameras and cell phones. The store makes a $100 profit on the sale of each digital camera x and a $50 profit on the sale of each cell phone y. The store wants to make a profit of at least $300 from its sales of digital cameras and cell phones. Write and graph an inequality that represents how many digital cameras and cell phones they must sell. Identify and interpret two solutions of the inequality.

Solution :

100x + 50y ≥ 300

50y ≥ -100 x + 300

y ≥ (-100/50) x + (300/50)

y ≥ -2x + 6

(1, 4) (2, 2) (3, 0) and (0, 6)

Example 4 :

Which ordered pair is not a solution of 

x + 5y < 15

a)  (-1, -3)     b)  (-1, 3)     c)  (1, 3)    d)  (3, 2)

Solution :

x + 5y < 15

Option a :

Applying the point (-1, -3) in the inequality given, we get

-1 + 5(-3) < 15

-1 - 15 < 15

-16 < 15

is a solution.

Option b :

Applying the point (-1, 3) in the inequality given, we get

-1 + 5(3) < 15

-1 + 15 < 15

14 < 15

is a solution.

Option c :

Applying the point (1, 3) in the inequality given, we get

1 + 5(3) < 15

1 + 15 < 15

16 < 15

is not a solution.

Option d :

Applying the point (3, 2) in the inequality given, we get

3 + 5(2) < 15

3 + 10 < 15

13 < 15

is a solution.

Option c is correct.

Example 5 :

The graph of which inequality is shown ?

a)  x + y ≤ -1       b) x + y ≥ -1   c)  x - y ≤ -1       b) x - y ≥ -1

Solution :

The shaded region is below the line,

slope = rise / run

= -1/1

m = -1

y-intercept = -1

y = mx + b

y = -1x - 1

Choosing one of the points from the shaded region (-1, -1)

-1 = -1(-1) - 1

-1 = 1 - 1

-1 = 0

Since it is solid line, it must be a less than or equal or greater than or equal sign.

x + y ≤ -1

Example 6 :

The inequality 9x + 5y ≥ 60 represents the number x of newspapers and the number y of magazines you must sell to earn enough points to earn a special school lunch. You sell four newspapers and six magazines. Do you receive a special school lunch? Explain.

Solution :

9x + 5y ≥ 60

x is the number of newspapers and y is the number of magazines.

x = 4 and y = 6

9(4) + 5(6) ≥ 60

36 + 54 ≥ 60

90 ≥ 60

Since the given point satisfies the inequality given, it must be the solution. Then you receive special school lunch.

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